note that the first set is compact, so is automatically uniformly continuous there, but as for the second set, for each is and this satisfies Lipschitz condition, hence we have uniform continuity on the second set, and finally on the whole set.

note that the first set is compact, so is automatically uniformly continuous there, but as for the second set, for each is and this satisfies Lipschitz condition, hence we have uniform continuity on the second set, and finally on the whole set.

You could also note that it's Lipschitz continuous on [1, infinity) since its derivative is bounded there.